Computational Mechanics

, Volume 55, Issue 6, pp 1071–1078

Vibration of structures containing compressible liquids with surface tension and sloshing effects. Reduced-order model

Original Paper

Abstract

This paper deals with the development of the linear vibration of a general viscoelastic structure, with a local wall acoustic impedance, containing an inviscid compressible liquid (but with an additional volume dissipative term), with surface tension (capillarity) and sloshing effects, and neglecting the effects of internal gravity waves and the elastogravity operator. The sloshing problems of incompressible liquids with capillarity effects in elastic structures exhibit a major difficulty induced by the boundary contact conditions on the triple line because the capillarity forces are forces per unit length while the elastic forces are forces per unit surface. The proposed framework has the following novel features: (i) introducing a new appropriate boundary condition for the contact angle condition compatible with a deformable structure considered here as viscoelastic, (ii) considering a compressible liquid while incompressibility hypothesis is generally used for FSI problems including capillarity phenomena, and (iii) constructing a reduced-order model for the computational coupled problem.

Keywords

Linear vibration Viscoelastic structure Surface tension Sloshing Contact angle condition Reduced-order-model 

References

  1. 1.
    Abramson HN (1966) The dynamical behaviour of liquids in moving containers. NASA SP-106Google Scholar
  2. 2.
    Amabili M, Paidoussis MP, Lakis AA (1998) Vibrations of partially filled cylindrical tanks with ring-stiffeners and flexible bottom. J Sound Vib 213(2):259–299CrossRefGoogle Scholar
  3. 3.
    Andrianarison O, Ohayon R (2006) Compressibiliy and gravity effects in internal fluid-structure vibrations: basic equations and appropriate variational formulations. Comput Methods Appl Mech Eng 195:1958–1972MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput Mech 43(1):3–37MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bauer HF, Eidel W (1990) Linear liquid oscillations in cylindrical container under zero-gravity. Appl Microgravity Technol 2:212–220 (ISSN 0931–9530)Google Scholar
  6. 6.
    Bazilevs Y, Takizawa K, Tezduyar TE (2013) Comput Fluid-Struct Interact. John Wiley, ChichesterCrossRefGoogle Scholar
  7. 7.
    Bermùdez A, Rodríguez R, Santamarina D (2003) Finite element computation of sloshing modes in containers with elastic baffle plates. Int J Numer Methods Eng 56(3):447–467MATHCrossRefGoogle Scholar
  8. 8.
    Cocciaro B, Faetti S, Nobili M (1991) Capillarity effects on the surface gravity waves in a cylindrical container: wetting boundary conditions. J Fluid Mech 231:325–343CrossRefGoogle Scholar
  9. 9.
    Concus P, Finn R (1969) On the behavior of a capillarity surface in a wedge. Proc Nat Acad Sci 63(2):292–299MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Cruchaga MA, Reinoso RS, Storti MA, Celentano DJ, Tezduyar TE (2013) Finite element computation and experimental validation of sloshing in rectangular tanks. Comput Mech 52:1301–1312MATHCrossRefGoogle Scholar
  11. 11.
    Das S, Marchand A, Andreotti B, Snoeijer JH (2011) Elastic deformation due to tangential capillary forces. Phys Fluids 23:072006CrossRefGoogle Scholar
  12. 12.
    Dias F, Kharif C (1999) Nonlinear gravity and capillary-gravity waves. Annu Rev Fluid Mech 31(1):301–346MathSciNetCrossRefGoogle Scholar
  13. 13.
    de Gennes PG (1985) Wetting: statics and dynamics. Rev Mod Phys 57:827–863CrossRefGoogle Scholar
  14. 14.
    de Gennes PG, Brochard-Wyart F, Quéré D (2004) Capillarity and wetting phenomena. Drops, bubbles, pearls, waves. Springer, New YorkMATHCrossRefGoogle Scholar
  15. 15.
    Dettmer W, Perić D (2006) A computational framework for free surface fluid flows accounting for surface tension. Comput Methods Appl Mech Eng 195:3038–3071MATHCrossRefGoogle Scholar
  16. 16.
    Dodge FT (2000) The new dynamical behaviour of liquids in moving containers. Southwest Research Institute, San AntonioGoogle Scholar
  17. 17.
    Dussan V, Ramé E, Garoff S (1991) On identifying the appropriate boundary conditions at moving contact line: an experimental investigation. J Fluid Mech 230:97–111CrossRefGoogle Scholar
  18. 18.
    El-Kamali M, Schotté JS, Ohayon R (2010) Computation of the equilibrium position of a liquid with surface tension inside a tank of complex geometry and extension to sloshing dynamic cases. Comput Mech 46:169–184MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    El-Kamali M, Schotté JS, Ohayon R (2011) Three-dimensional modal analysis of sloshing under surface tension. Int J Numer Methods Fluids 65:87–105MATHCrossRefGoogle Scholar
  20. 20.
    Faltinsen OM, Timokha AN (2009) Sloshing. Cambridge University Press, CambridgeGoogle Scholar
  21. 21.
    Farhat C, Chiu EKY, Amsallem D, Schotté JS, Ohayon R (2013) Modeling of fuel sloshing and its physical effects on flutter. AIAA J 51(9):2252–2265CrossRefGoogle Scholar
  22. 22.
    Felippa CA, Park KC, Farhat C (2001) Partitioned analysis of coupled mechanical systems. Comput Methods Appl Mech Eng 190:3247–3270MATHCrossRefGoogle Scholar
  23. 23.
    Felippa CA, Park KC, Ross MR (2010) A classification of interface treatments for FSI. Fluid structure interaction II. Springer, BerlinGoogle Scholar
  24. 24.
    Finn R (2001) On the equations of capillary. J Math Fluid Mech 3:139–151MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Finn R (2006) The contact angle in capillarity. Phys Fluids 18:047102MathSciNetCrossRefGoogle Scholar
  26. 26.
    Finn R, Luli GK (2007) On the capillary problem for compressible fluids. J Math Fluid Mech 9:87–103MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Gonzàlez JA, Park KC, Lee I, Felippa CA, Ohayon R (2012) Partitioned vibration analysis of internal fluid-structure interaction problems. Int J Numer Methods Eng 92(3):268–300CrossRefGoogle Scholar
  28. 28.
    Harari I, Grosh K, Hughes TJR, Malhotra M, Pinsky PM, Stewart JR, Thompson LL (1996) Recent development in finite element methods for structural acoustics. Arch Comput Methods Eng 3(2–3):131–309MathSciNetCrossRefGoogle Scholar
  29. 29.
    Henderson DM, Miles JW (1994) Surface-wave damping in a circular cylinder with a fixed contact line. J Fluid Mech 275:285–299Google Scholar
  30. 30.
    Herczynski A, Weidman PD (2012) Experiments on the periodic oscillation of free containers driven by liquid sloshing. J Fluid Mech 693:216–242MATHCrossRefGoogle Scholar
  31. 31.
    Ibrahim R (2005) Liquid sloshing dynamics: theory and applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  32. 32.
    Keller JB, Merchant G (1991) Flexural rigidity of a liquid surface. J Stat Phys 63:1039–1051MathSciNetCrossRefGoogle Scholar
  33. 33.
    Landau L, Lifchitz E (1992) Fluid mechanics. Pergamon Press, OxfordGoogle Scholar
  34. 34.
    Lighthill J (1978) Waves in fluids. Cambridge University Press, CambridgeMATHGoogle Scholar
  35. 35.
    Miras T, Schotté JS, Ohayon R (2012) Energy approach for static and linearized dynamic studies of elastic structures containing imcompressible liquids with capillarity: a theoretical formulation. Comput Mech 50:729–741MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Moiseyev NM, Rumyantsev VV (1968) Dynamics stability of bodies containing fluid. Applied physics and engineering, vol 6. Springer, New YorkCrossRefGoogle Scholar
  37. 37.
    Morand HJP, Ohayon R (1995) Fluid structure interaction. John Wiley, ChichesterMATHGoogle Scholar
  38. 38.
    Myshkis AD, Babskii VG, Kopachevskii ND, Slobozhanin LA, Tyuptsov AD, Wadhwa RS (1987) Low-gravity fluid mechanics. Springer, BerlinCrossRefGoogle Scholar
  39. 39.
    Ohayon R (2004) Reduced models for fluid-structure interaction problems. Int J Numer Methods Eng 60(1):139–152MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Ohayon R, Soize C (1998) Structural acoustics and vibration. Academic Press, LondonGoogle Scholar
  41. 41.
    Ohayon R, Soize C (2012) Advanced computational dissipative structural acoustics and fluid-structure interaction in low- and medium-frequency domains. Reduced-order models and uncertainty quantification. Int J Aeronaut Space Sci 13(2):127–153Google Scholar
  42. 42.
    Ohayon R, Soize C (2014) Advanced computational vibroacoustics: reduced-order models and uncertainty quantification. Cambridge University Press, New YorkCrossRefGoogle Scholar
  43. 43.
    Ostrach S (1982) Low-gravity fluid flows. Annu Rev Fluid Mech 14:313–345CrossRefGoogle Scholar
  44. 44.
    Poincaré H (1895) Capillarité (Capillarity). Georges Carré, Editeur, Paris (reprinted by Editions Jacques Gabay, Paris, p 2006Google Scholar
  45. 45.
    Pukhnachev VV, Solonnikov VA (1982) On the problem of dynamic contact angle. J Appl Math Mech 46:961–971Google Scholar
  46. 46.
    Reynolds WC, Satterlee HM (1966) Liquid propellant behavior at low and zero g. In: “The dynamical behaviour of liquids in moving containers”, Abramson HN editor, NASA SP-106Google Scholar
  47. 47.
    Schotté JS, Ohayon R (2005) Incompressible hydroelastic vibrations: finite element modelling of the elastogravity operator. Comput Struct 83:209–219CrossRefGoogle Scholar
  48. 48.
    Schotté JS, Ohayon R (2013) Linearized formulation for fluid-structure interaction: application to the linear dynamic response of a pressurized elastic structure containing a fluid with a free surface. J Sound Vib 332:2396–2414CrossRefGoogle Scholar
  49. 49.
    Shankar PN, Kidambi R (2005) The contact angle in inviscid fluid mechanics. Proc Indian Acad Sci (Math. Sci.) 115(2):227–240MathSciNetCrossRefGoogle Scholar
  50. 50.
    Shikhmurzaev YD (1997) Moving contact lines in liquid/liquid: solid systems. J Fluid Mech 334:211–249MATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Starov VM (1992) Equilibrium and hysteresis contact angles. Adv Colloid Interface Sci 39:147–173CrossRefGoogle Scholar
  52. 52.
    Takizawa K, Tezduyar TE (2011) Multiscale space-time fluid-structure interaction techniques. Comput Mech 48:247–267MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Tezduyar T (2006) Interface-tracking and interface-capturing techniques for finite element computation of moving boundaries and interfaces. Comput Methods Appl Mech Eng 195:2983–3000MATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    Tezduyar T, Behr M, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces: the deforming-spatial-domain/space-time procedure: I. The concept and preliminary numerical tests. Comput Methods Appl Mech Eng 94:339–351MATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    Tezduyar T, Behr M, Mittal S, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces: the deforming-spatial-domain/space-time procedure: II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders. Comput Methods Appl Mech Eng 94:353–371MATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Thompson PA, Robbins MO (1989) Simulation of contact line motion: slip and the dynamic contact angle. Phys Rev Lett 63:766–769Google Scholar
  57. 57.
    Veldman AEP, Gerrits J, Luppes R, Helder JA, Vreeburg JPB (2007) The numerical simulation of liquid sloshing on board spacecraft. J Comput Phys 224:82–99MATHMathSciNetCrossRefGoogle Scholar
  58. 58.
    White LR (2003) The contact angle on an elastic substrat. I. The role of disjoining pressure in a surface interface. J Colloid Interface Sci 258:82–96CrossRefGoogle Scholar
  59. 59.
    Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics, vol 6. Elsevier, Butterworth-Heinemann, AmsterdamMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Structural Mechanics and Coupled Systems LaboratoryConservatoire National des Arts et Métiers (CNAM)ParisFrance
  2. 2.Modelisation et Simulation Multi-Echelle, MSME UMR 8208 CNRSUniversité Paris-EstMarne-la-Vallée Cedex 02France

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